Hydrogen-like atom

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Hydrogen-like atoms (or hydrogenic atoms) are atoms with one single electron. Like the hydrogen atom, hydrogen-like atoms are one of the few quantum mechanical problems which can be exactly solved. Atoms or ions whose valence shell is made of one single electron (e.g. alkali metals) have similar chemical bonding or spectroscopic propreties to hydrogen-like atoms.

The simplest atomic orbitals are those that occur in an atom with a single electron, such as the hydrogen atom. In this case the atomic orbitals are the eigenstates of the hydrogen Hamiltonian. They can be obtained analytically (see Hydrogen atom). An atom of any other element ionized down to a single electron is very similar to hydrogen, and the orbitals take the same form.

For atoms with two or more electrons, the governing equations can only be solved with the use of methods of iterative approximation. Orbitals of multi-electron atoms are qualitatively similar to those of hydrogen, and in the simplest models, they are taken to have the same form. For more rigorous and precise analysis, numerical approximations must be used. Atomic orbitals are often expanded in a basis set of Slater-type orbitals which are orbitals of hydrogen-like atoms with arbitrary nuclear charge Z.

A given (hydrogen-like) atomic orbital is identified by unique values of three quantum numbers: n, l, and m_l. The rules restricting the values of the quantum numbers, and their energies (see below), explain the electron configuration of the atoms and the periodic table.

The stationary states (quantum states) of the hydrogen-like atoms are its atomic orbital. However, in general, an electron's behavior is not fully described by a single orbital. Electron states are best represented by time-depending "mixtures" (linear combinations) of multiple orbitals. See Linear combination of atomic orbitals molecular orbital method.

The quantum number n first appeared in the Bohr model. It determines, among other things, the distance of the electron from the nucleus; all electrons with the same value of n lay at the same distance. Modern quantum mechanics confirms that these orbitals are closely related. For this reason, orbitals with the same value of n are said to comprise a "shell". Orbitals with the same value of n and also the same value of l are even more closely related, and are said to comprise a "subshell".

1 Mathematical characterization
2 See also
3 References

[edit] Mathematical characterization

[edit] Derivation

Atomic orbitals are solutions to the Schrödinger equation. In this case, the potential term is the potential given by Coulomb's law:

$V = -\frac{1}{4 \pi \epsilon_0} \frac{Ze^2}{r}$

where

The first term is a constant, usually abbreviated by the letter k,
Z is the atomic number,
e is the elementary charge,
r is the magnitude of the distance from the nucleus.

The wavefunction is a function of three spatial variables, so that after removing the time-dependence, the Schrödinger equation is a partial differential equation in three variables (see separation of variables). However, since the potential is spherically symmetric, it is profitable to write the equation in spherical coordinates. In this form, any individual eigenfunction ψ can be written as a product of three single-variable functions, often denoted as follows:

ψ(r,θ,φ) = R (r) f (θ) g (φ)

(where θ represents the polar angle (colatitude) and φ the azimuthal angle.) It can further be reduced to three separate equations, each in one variable.

Two separations are required, resulting in two separation constants. A third arbitrary constant results from the application of boundary conditions to R. The equations given below use a form of the separation constants that seems arbitrary, but it simplifies matters later on.

$\frac{1}{R(r)} \frac{d}{dr} \left ( r^2 \frac{dR}{dr} \right ) + \frac{2 \mu r^2}{\hbar^2}(E-V(r)) = l(l+1)$

$\frac{1}{g(\phi)} \frac{d^2 g(\phi)}{d\phi^2} = -m^2$

$l(l+1)\sin ^2(\theta) + \frac{\sin(\theta)}{f(\theta)} \frac{d}{d\theta} \left [ \sin(\theta) \frac{df}{d\theta} \right ] = m^2$

where:

ħ is the reduced Planck constant ( $\frac{h}{2\pi}$ ), and
μ is the reduced mass of the electron vis-à-vis the nucleus.

[edit] Results

In addition to $\ell$ and m, a third arbitrary integer, called n, emerges from the boundary conditions placed on R. The functions R, f and g that solve the equations above depend on the values of these integers, called quantum numbers. As a result, it is customary to subscript the functions with the values of the quantum numbers they depend on. The forms of the functions are:

$\psi = C_{nlm}\, R_{nl}(r)\, f_{lm}(\theta)\, g_m(\phi)$

$R_{nl}(r) = e^{-\frac{Z r}{n a_{\mu}}} \left ( \frac{2 Z r}{n a_{\mu}} \right )^l L_{n-l-1}^{2l+1} \left( \frac{2 Z r}{n a_{\mu}} \right)$

$f_{lm}(\theta) = \frac{(\sin\theta)^{|m|}}{2^l l!} \left [ \frac{d}{d(\cos\theta)} \right ]^{l+|m|}(\cos ^2(\theta)-1)^l \qquad$ (the associated Legendre functions)

g m (φ) = e i m φ

$n, l, m$ are integer numbers, with the restrictions:

n = 1,2,3,...

l = 0,1,2,..., n - 1

m = - l, - (l - 1),...,0,..., l - 1, l

where:

$L_{n-l-1}^{2l+1}$ are the generalized Laguerre polynomials.
$a μ$ is defined by:

$a_{\mu} = {{4\pi\varepsilon_0\hbar^2}\over{\mu e^2}}$

Note that

a μ

is approximately equal to

a 0

(the Bohr radius)

i is the imaginary number.

$C n l m$ is a normalization constant. Since the wavefunction must be normalized ( $\int_{{\Bbb{R}}^3} | \Psi |^2 d^3\vec{r} = 1$ ), by calculating this integral we get:

$C_{nlm} = \left [ \left ( \frac{2 Z}{n a_{\mu}} \right )^3 \frac{(n - l - 1)!}{2n (n+l)! } \right ]^{1/2}$

The functions f and g are usually consolidated into the function $Y (θ,φ) = f (θ) g (φ)$ . This function is a spherical harmonic.

Thus, the complete expression for the normalized wavefunctions is:

$\psi_{nlm}(\theta,\phi,r) = \sqrt {{\left ( \frac{2 Z}{n a_{\mu}} \right ) }^3\frac{(n-l-1)!}{2n[(n+l)!]} } e^{- Z r / {n a_{\mu}}} \left ( \frac{2 Z r}{n a_{\mu}} \right )^{l} L_{n-l-1}^{2l+1} \left ( \frac{2 Z r}{n a_{\mu}} \right ) \cdot Y_{l,m}(\theta, \phi )$

[edit] Angular momentum

Each atomic orbital is associated with an angular momentum L. It is a vector, and its magnitude is given by:

$\left| \mathbf L \right| = \hbar \sqrt{l(l+1)}$

The projection of this vector onto any arbitrary direction is quantized. If the arbitrary direction is called z, the quantization is given by:

$L_z = m_l\hbar$

where $m l$ is restricted as described above. This value is always less than the total angular momentum. Thus, if the $\mathbf L$ -vector is measured in some direction, it will not lie entirely in that direction; part of it will lie in perpendicular directions. This allows the uncertainty principle to stand. It mandates that no two components of $\mathbf L$ may be known at once. If one component were known to be equal to the total $\mathbf L$ , the other two would necessarily be zero.